Hass, Joel; Nowik, Tahl Invariants of knot diagrams. (English) Zbl 1161.57002 Math. Ann. 342, No. 1, 125-137 (2008). The authors define a new family of knot diagram invariants of order one. These invariants are derived from invariants of two component links. Note that an invariant on knot diagrams is said to be of order one, if whenever we simultaneously perform two Reidemeister moves on a diagram, in two disjoint discs \(A, B \subset S^2\), then the change in the invariant due to the move in \(A\) is not affected by whether we first perform the move in \(B\). As an application, the authors determine the minimal number of Reidemeister moves between two specific diagrams of the unknot with \(2n+1\) crossings. The minimal number is \(2n+2\). Reviewer: Youngsik Huh (Seoul) Cited in 1 ReviewCited in 10 Documents MSC: 57M25 Knots and links in the \(3\)-sphere (MSC2010) Keywords:Reidemeister move; knot diagram; invariant of diagrams × Cite Format Result Cite Review PDF Full Text: DOI arXiv References: [1] Alexander, J.W., Briggs, G.B.: Ann. Math. 28, 562–586 (1926/27) · JFM 53.0549.02 · doi:10.2307/1968399 [2] Arnold V.I. (1994). Adv. Soviet Math. 21: 33–91 [3] Carter, J.S., Elhamdadi, M., Saito, M., Satoh, S.: A lower bound for the number of Reidemeister moves of type III. Topology Appl. 153, 2788–2794 (2006) · Zbl 1106.57003 · doi:10.1016/j.topol.2005.11.011 [4] Chmutov, S., Duzhin, S.: In: Arnold-Gelfand Mathematical Seminars: Geometry and Singularity Theory, pp. 123–138. Birkhauser, Boston (1997) [5] Fiedler T. (2001). Gauss Diagram Invariants for Knots and Links. Kluwer, Dordrecht · Zbl 1009.57001 [6] Hagge T.J. (2006). Proc. AMS 134: 295–302 · Zbl 1078.57007 · doi:10.1090/S0002-9939-05-07935-9 [7] Hass J. and Lagarias J.C. (2001). J. Am. Math. Soc. 14: 399–428 · Zbl 0964.57005 · doi:10.1090/S0894-0347-01-00358-7 [8] Hass, J., Nowik, T.: arXiv:0711.2350 [9] Hayashi C. (2006). J. Knot Theory Ramifications 15: 313–325 · Zbl 1094.57007 · doi:10.1142/S0218216506004488 [10] Kauffman L.H. (1987). On Knots. Princeton University Press, Princeton · Zbl 0627.57002 [11] Östlund, O.: J. Knot Theory Ramifications 10, 1215–1227 (2001) · Zbl 0998.57021 · doi:10.1142/S0218216501001402 [12] Polyak, M.: Topology 37, 989–1009 (1998) · Zbl 0961.57015 · doi:10.1016/S0040-9383(97)00013-X [13] Polyak M. and Viro O. (2001). J. Knot Theory Ramifications 10: 711–738 · Zbl 0997.57021 · doi:10.1142/S0218216501001116 [14] Reidemeister H. (1926). Abh. Math. Sem. Univ. Hamburg 5: 7–23 · JFM 52.0578.04 · doi:10.1007/BF02952506 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.